OPTIMAL RESTRICTED DESIGNS FOR THE INVERSE GAUSSIAN MEAN

This study deals with the development of optimal restricted two and three stage designs when response variable has an inverse Gaussian distribution with known scale parameter. Key WordsOptimal restricted two and three-stage design, Inverse Gaussian distribution


INTRODUCTION
Fixed sample size design is not useful in experiment that subjects enter to the study sequentially.Consequently it is possible to analyze the accumulated data sequentially.Wald [1], introduced sequential analysis and demonstrating that sequential probability ratio test (SPRT) require substantially fewer observations than a fixed sample test of equivalent statistical power.
SPRT is widely used in clinical trials, quality control studies and life tests.Also generally sequential designs cannot be used in such situations.In this case analyzing the accumulated data in groups is the most convenient way.
Data are analyzed after groups of observations are entered into a group sequential design.However group sequential designs are generally more practical and they provide much of the saving possible from sequential designs [2].
Group sequential designs are widely used in clinical trials.In most randomized clinical trials with sequential patient entry, fixed sample size design is unjustified on ethical grounds and sequential designs are often impractical.Two and three stage design is the simplest form of a group sequential design.Case et al. [3], [4], developed optimal restricted two (OR 2 ) and three (OR 3 ) stage design that have the restriction of using the fixed sample critical value at the final stage.
In general, optimal restricted two and three-stage designs has been proposed for normally and binomial response variable.In this study, a optimal restricted designs when response variable has an inverse Gaussian distribution with known scale parameter is proposed.Inverse Gaussian distribution function is a very useful alternative to the real life time distribution such as gamma, log-normal and Weibull distributions.The distribution has a wide application area in clinical trials, quality and reliability theory, industrial engineering applications and life tests.In those areas, outcome variable can be measured in series because data is accumulated sequentially.[5,6] Edgeman and Salzberg [6] and Edgeman and Lin [7] developed the sequential probability ratio test for the inverse Gaussian mean, and its application to sequential sampling plans.Bacanlı and Demirhan [8] suggested the group sequential test when response variable has an inverse Gaussian distribution with known scale parameter.
This study is organized as follows: In section 2, the optimal restricted designs are described.In section 3, SPRT for the mean of inverse Gaussian distribution is briefly reviewed and it is shown that optimal restricted designs can be used in inverse Gaussian mean.Example and the optimal restricted designs comparison to other design are given in section 4.

OPTIMAL RESTRICTED DESIGNS
In this section, firstly OR 2 design is examined for response variable has normal distribution with mean (θ) and known variance (σ 2 ).For testing where  ˆ is computed from data on all n observations.If 3 C Z  ; Accept 0 H , otherwise, reject 0 H [2]. Z 1 and Z are distributed standard normal distribution and their joint distribution is bivariate normal with zero means, unit variances, and correlation (n 1 /n) 1/2 .The maximum sample size for the two-stage design is n and is realized whenever a second stage is necessary.The expected sample size (ESS) of the two-stage design is given by equation (3): where    S P denote the probability that the trial will be stopped at the first stage, and p is the rate of the number of observations at the first stage to the number of total observations at the second stage n n p 1


. For some studies it might be practical to choose equal samples at each stage.Therefore, if p=0.50, each stage have equal sizes. S. Bacanlı value can be computed for There are five unknown parameters in the two-stage design, namely: and   3 C .The critical value at the second stage, 3 C , will be set to equal that of the fixed sample test denotes the standard normal distribution function.The other four parameters of interest are chosen to satisfy the two equations: where, and 5) and ( 6) are solved iteratively by numerical integration of the bivariate normal distribution using a double precision function [3,9].
With five parameters and only three constraints given by equations ( 4), ( 5), ( 6) optimality criteria are used to determine the parameter values.So, this test is called optimal restricted two-stage design.In this study, we have examined Bayes criteria.
Bayes Criterion: Minimize a weighted average of the ESS under 0 H and the Using a weight of 0 for this criterion gives the most efficient designs if the null hypothesis is true while a weight of 1 gives the most efficient designs if the specified alternative is true .
The optimal design parameters, the probabilities p s (), maximum (n) and expected sample sizes (ESS) obtained using the bayes criteria are listed in Table 1  .In tables, f n is the sample size for a fixed sample design [3,9].
The sample size required for a OR 2 design is obtained by multiplying the tabled values by 2   =/( 1 - 0 ) 2 OR 3 design is an extension of the OR 2 design to three stages.However the sample sizes must be equal for each stage of the design [4].
The OR 3 design for normal mean testing is given as follows: Stage I: Accure n 1 observations and calculate test statistics , Where  ˆ is calculated from data on the first n 1 observation.If where  ˆ is computed from data on all n observations.If where  ˆ is computed from data on all n observations.If With six parameters and only two constraints, parameter values are chosen to min ESS() for Bayes criteria.Therefore the algorithm used to obtain the parameter values for OR 3 design is olmost identical in the OR 2 [4,9].
The design parameters and the sample sizes obtained using Bayes criteria for OR 3 design are given in Table 2

OPTIMAL RESTRICTED DESIGNS FOR THE MEAN OF AN INVERSE GAUSSIAN DISTRIBUTION
Let x is an inverse Gaussian (IG) distributed random variable and its probability density function is defined as follows; Here  is the mean of the distribution.So it is a location parameter and  is a scale parameter [5].
Given a sequence of observations  , , x from inverse Gaussian distribution (12), suppose one wishes to test the simple null hypothesis H 0 : 0 against the simple alternative H 1 :  1 ( 1   0 ) , when λ is known.The SPRT for testing H 0 is defined as follows: IG distribution is related to normal distribution.This relation was given with following theorem in Chikara and Folks [5], which establishes a basic relationship between IG and the normal.
. Then the pdf of Y is given by is one-to-one and as x varies from 0 to  , y varies from   to  .
Then the cumulative distribution function   Because of this and because of the one-to-one relationship between x and y , one finds that the distribution of X is asymptotically normal with mean  and variance   / 3 [5].Therefore fixed sample test for the mean of an inverse Gaussian distribution based on the standard normal distribution.
Given a random sample x 1 , x 2, ,…, x n drawn from IG distribution.Consider the testing of the hypothesis, , when  is known, the test statistic for the mean is defined as,   Here, . The test statistic is compared with where  denotes the standard normal distribution function.Consequently if H is rejected for two sided hypothesis [5,8].
In the sense of this information, we modify restricted optimal two-stage design for the mean of an inverse Gaussian distribution.Test statistics of OR 2 can be defined from equation ( 1), (2), and (17), Stage I: Accrue 1 n observations and calculate test statistic,   Where x is calculated from data on the first n 1 observation.
Stage II: Accrue additional 2 n observations.Let Where x is calculated from data on the first n 1 observation.
According to Theorem 1, Z 1 and Z have a standard normal distribution.Therefore, it is suggested that the OR 2 and OR 3 designs can be used for testing inverse Gaussian distribution mean with known scale parameter.
OR 3 design is an extension of OR 2 design to three stages and test statistics Z i can be obtained as given OR 2 design.
In this case, design parameters for OR 2 and OR 3 designs are the same as normal distribution parameters.However, the sample size is obtained by multiplying the values . Therefore, the only change in design is the sample size.

COMPARISON WITH OTHER DESIGN AND DISCUSSION
In this section, the comparison of optimal restricted designs (OR 2 and OR 3 ) with fixed sample design and SPRT for inverse Gaussian distribution have been examined with an example.
As an example, suppose It is well known that the SPRT has the minimum ESS(H 1 ), but it hasn't got a finite maximum number of observation.Furthermore OR 3 design needs a smaller sample size than other designs.In this study, ıt is seen that these results are also valid for inverse Gaussian distribution [1,4].
Let S SPRT ( 1 ) = n f -ESS SPRT ( 1 ) i=0,1 denote the savings possible with the use of the SPRT design.Also S 2 ( 1 ) = n f -ESS 2 ( 1 ) and S 3 ( 1 ) = n f -ESS 3 ( 1 ) denote the savings possible with optimal restricted two and three stage designs.Then the ratios R give the proportion of the possible savings realized with two and three-stage designs.A comparison of sequential design and the optimal restricted design according to proportion of the possible savings is given Table 3.
In Table 3, it is clear that OR 2 design can provide approximately 50% of the savings that would have been realized with SPRT design.OR 3 design provides as much as 60% of the possible savings.
Other examples with different choices of , , , and  could readily be presented to illustrate the same general principle.
Case at all [4], compared those designs for the mean of normal distribution and obtained similar results.Therefore, as for normal distribution, it is advantageous to use optimal restricted two and three-stage design for the mean of inverse Gaussian distribution.

H
, otherwise; continue the second stage.

Stage 2 : 2 n
Accrue an additional observations.Let unknown parameters in the OR 3 design, namely n 1 , n 2 , n 3 ,C 1 , C 2 , C 3 , C 4 , and C 5 .The critical value at the final stage, will be set equal to that of the fixed sample test.OR 3 design considers the case of equal sample sizes at each stage, reducing the number of unknown parameters to six.

Table 1 .
Optimal restricted two-stage one-sided designs for bayes criterion

Table 3 .
Comparison of fixed, SPRT and OR 2 , OR 3 designs